Methods and Applications of Analysis © 1998 International Press 5.(4) 1998, pp. 439-452 ISSN 1073-2772 ON DIFFERENTIAL EQUATIONS FOR ORTHOGONAL POLYNOMIALS Mourad E. H. Ismail and Jet Wimp ABSTRACT. We find four linear independent solutions of the fourth-order differ- ential equation satisfied by the associated Jacobi polynomials. We show that the coefficients in the lowering operator for general orthogonal polynomials sat- isfy inhomogeneous four-term recurrence relations and derive further properties of them. In addition, we show that the associated polynomials {Pn (x)} for pos- itive integers c satisfy a linear differential equation of order four, we identify a basis of solutions of the differential equation, and we establish similar results for co-recursive polynomials. 1. Introduction Let {pn(%)} be orthonormal with respect to a weight function w supported on an interval [a, 6], finite or infinite, that is / 'Pm(x)p n (x)w(x) dx = 6 m% n. (1.1) Ja With the weight function w we associate an external field v through w(x)=e- v ( x \ (1.2) The p n s satisfy a three-term recurrence relation xpn(x) = a n+1 p n+1 (x) + b n p n (x) + a n p n _i(aO, n > 0, (1.3) with Po(x) = 1, pi(aO = (x - bo)/ai. (1.4) We shall assume ^ ( t:; ,(y W -0,1,..., (i.5) Received January 5, 1998, revised September 23, 1998. 1991 Mathematics Subject Classification: Primary 42C05, secondary 33C45. Key words and phrases: orthogonal polynomials, functions of the second kind, recurrence rela- tions, differential equations, associated polynomials, co-recursive polynomials. 439